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Treating G(B→lnXc) with the OPE: Presenting it the 'Royal' Way

This text presents a method for treating the decay process G(B→lnXc) using the Operator Product Expansion (OPE). It discusses the use of Heavy Quark Parameters (HQP) and energy/mass moments to determine the HQP with high precision. The impact of experimental cuts on the analysis is also explored. The study emphasizes the importance of keeping the cuts as low as possible to minimize bias in the measured moments and to address systematic uncertainties.

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Treating G(B→lnXc) with the OPE: Presenting it the 'Royal' Way

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  1. SLAC 12/’03 TreatingG(BÆln Xc) with the OPE--Presenting it the “Royal” Way Ikaros Bigi Notre Dame du Lac Wilsonian cut-off scale ~ 1 GeV G(HQÆln Xq)= GF2|KM|2mQ5( m ) /192p3 • HQP€ moments • constraints • systematicuncert. These results are not classified

  2. First Step: Width in terms of HQP Benson,Mannel, Uraltsev,IB, NuPhy. B665,367 G(BÆln Xc) =F(V(cb),HQP:mQ, mp2,…) ±1-2%,th. limiting factor: perturb. correct. to nonpert. contrib. • Caveat: do not rely on expansion in 1/mch! • do not impose constraint a priori mb-mc= <MB> - <MD>+mp2(1/2mch-1/2mb)+nonlocal op. can check it a posteriori • all order BLM and second order non-BLM contrib. to bb • nonpert. contrib. through order 1/mQ3 • including intrinsic charm (b…c)(c…b)

  3. mb,kin(m) €MS mass mb(mb) Benson,Mannel, Uraltsev,IB, NuPhy. B665,367

  4. pole mass: intrinsic uncertainty ~ LQCD • need short distance mass • MS mass mb(m): m = mb “unnaturally” high scale m << mb - diverges for m Æ 0 • prefer `low-scale running’ mass • ‘kinetic’ mass Voloshin SR dmb,kin(m)/dm= -(16/9)(aS (m)/p) -(16/9)(aS (m)/p)(m/mb) +…

  5. HQ Sum Rules • r2(m) - 1/4 = Sn |t1/2(n) |2 + 2 Sm |t3/2(m) |2 Bj 1990 • 1/2= - 2 Sn |t1/2(n) |2 + Sm |t3/2(m) |2 U 2000 • L(m) = 2 (Snen |t1/2(n) |2 + 2 Smem |t3/2(m) |2) Vo 1992 • m2p(m)/3=Snen2 |t1/2(n) |2 + 2 Smem2|t3/2(m) |2 BiSUVa 1994 • m2G(m)/3=-2Snen2 |t1/2(n) |2 + 2Smem2|t3/2(m) |2 BSU 1997 • r3D(m)/3=Snen3 |t1/2(n) |2 + 2Smem3|t3/2(m) |2ChPir 1994 • -r3LS(m)/3=-Snen3 |t1/2(n) |2 + 2Smem3|t3/2(m) |2BSU 1997 where:t1/2&t3/2denote transition amplitudes for BÆlnD(sq = 1/2 or 3/2)with excitation energyek£m • rigorous definitions, inequalities +experim. constraints

  6. BSUV, Phys.Rev. D56 (1997) 4017 … A. Hoang, hep-ph/0204299 M. Battaglia et al., hep-ph/0304132

  7. Second Step: Determining HQP On the power of the OPE • a host of observables expressed by a universal cast of Heavy Quark Parameters (=expect. values of local operators): mQ,kin, mp2, mG2, rD3, rLS3, … memento: only2local operators in O(1/mQ3 ) ! mQ,kin, mp2, mG2, …€mQ,PS, l1, l2,…: Benson et al., NuPhy. B665,367 CKM Unitarity Triangle WS Proceed., hep-ph/0304132 • energy & mass moments yield HQP to be used universally • caveat: in general not a 1-to-1 correspondence moments €HQP • need linear combinations of moments

  8. |V(cb)/0.042| = 1-0.66[mb- 4.6 GeV] + 0.39(mc-1.15 GeV) + 0.05(mG2-0.35 GeV2) + 0.013(mp2-0.4 GeV2) + 0.09(rD3-0.2 GeV3)- 0.01 (rLS3+0.15 GeV3) energy/had. mass momentsÆHQP M1(El) =G-1ÚdEl El dG/d El Mn(El) =G-1ÚdEl[El- M1(El)]ndG/d El , n > 1 M1(MX) =G-1ÚdMX2 (MX2- MD2) dG/dMX2 Mn(MX) =G-1ÚdMX2 (MX2- <MX2>)ndG/dMX2 , n > 1 `ultimately’ can determine rLS3 [& mG2 in principle] yet for now -- at least as an option -- ‘seed’mG2 & rLS3 mG2 = 0.35 GeV2, rLS3 = - 0.12 GeV 2

  9. |V(cb)/0.042| = 1-0.66[mb- 4.6 GeV] + 0.39(mc-1.15 GeV) + 0.05(mG2-0.35 GeV2) + 0.013(mp2-0.4 GeV2) + 0.09(rD3-0.2 GeV3)- 0.01 (rLS3+0.15 GeV3) |Vcb|=0.0416¥(1±0.017|exp±0.015|G(B)±0.015|HQP) Achille “us” vs. “dmb ~ 2% implyingd|Vcb| > 5%”??? low moments depend on ~ same comb. of HQP! [.=F(mb-0.65mc)] [Gpartµmb2(mb-mc)3] |V(cb)/0.042| = 1- 1.70 [<El>-1.38 GeV]- 0.075 (mc-1.15 GeV) - 0.085(mG2-0.35 GeV2) + 0.07(mp2-0.4 GeV2) - 0.055(rD3-0.2 GeV3)-0.005 (r LS3+0.15 GeV3)

  10. M1-3(El) & M1(MX) & G(BÆln Xc) = F(mb-0.65mc) • facilitates analysis for V(cb) • complicates it for V(ub) -- G(BÆln Xu) = G(mb) -- yet only as a matter of practice, not of principle! M2,3(MX) exhibit different dependence • need higher mass moments

  11. Impact of experimental cuts • Experimental cuts on energyetc. applied for practical reasons • yet they degrade -- even `corrode’ -- `hardness’ Q of transition • `exponential’ contributions exp[-cQ/mhad]missed in usual OPE expressions • quite irrelevant for Q >> mhad • yet relevant for Q ~ mhad! for B Æg Xq: Q = mb - 2 Ecut e.g.: for Ecut~ 2 GeV, Q ~ 1 GeV !

  12. earlier work by C. Bauer • considers verydifferent effects • addresses theor. uncertaint., notbiases! Pilot study (Uraltsev, IB) ~ 1.5 % shift for Ecut=2 GeV ~ 40 % shift for Ecut=2 GeV absolute bias due to experim. cut 2 different ansaetze for distribution function [curves shown for mb=4.6 GeV, mp2 = 0.45 GeV2; bias depends on HQP]

  13. terms ~ O(1/mQ3) irrelevant for this analysis • only 3 dimensional parameters: mb, mp2, Q = mb - 2 Ecut • simplescaling behaviour arises Don Benson more refined studies: at Ecut = 2 GeV mb shifted by ~ 50 MeV mp2 shifted by ~ 0.1 GeV2

  14. hadronic mass moments in B Æl n Xc • encouraging Lessons: • keep the cuts as low as possible • biasin the meas. moments induced by cuts • can be corrected for • not a pretext for inflating theor. uncert. • moments meas. as fction of cuts: important cross check!

  15. Quality control & systematic uncertainties almost alltheoretical uncertainties systematic in nature: • unknown higher order perturb. & nonperturb. contrib. • limitations to quark-had duality Overconstraints most powerful protection against ignorance • Lenin’s dictum: “Trust is good -- control is better!” • measure higher (2nd and 3rd) moments • of different types

  16. M1-3(El) & M1(MX) yield consistent values for mb-0.65mc • M1-3(El) & M1,2[,3](MX) yield consistent values for mb,mc • value of mb thus obtained consistent with value inferred from Y(4S) Æ bb • keep energy cuts as low as possible • yet analyze moments as function of (reasonable) cuts • such overconstraints provide you with measure of “sufficient inclusiveness” case study: study moment analysis as function of P* my guess: inconsistencies emerge when P*Æ 1.6 - 1.7 GeV

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